Day 8: Two Step Equation Errors

Quote of the Day: “In one of the most famous experiments in the history of psychology, Walter Mischel and his students exposed four-year-old children to a cruel dilemma. They were given a choice between a small reward (one Oreo), which they could have at any time, or a larger reward (two cookies) for which they had to wait 15 minutes under difficult conditions. They were to remain in the room alone, facing a desk with two objects a single cookie and a bell that the child could ring at any time to call in the experimenter and receive the one cookie. Ten or fifteen years later, a large gap had opened between those who had resisted temptation and those who had not. The resisters had higher measures of executive control in cognitive tasks, and especially the ability to reallocate their attention effectively. As young adults, they were less likely to take drugs. A significant difference in intellectual aptitude emerged: the children who had shown more self-control as four-year-olds had substantially higher scores on tests of intelligence.” – Daniel Kahneman

Question of the Day (from a Student): Why is it that there is a 6 out of nowhere when five is divided by six?

Lesson Objective (Regular Math): Convert repeating decimals into fractions

Lesson Sequence: I read the quote and question first.

Next was the Number Talk. The number talks have gotten better everyday, but this was to be expected. I would like to see more students participate because everyday for the past four days now has been a subtraction problem. As I said there have been improvements, so students are no longer simply using the standard algorithm. My colleague and I discussed the possibility of doing a reflection. We could get the students to answer the question how many times have I volunteered to speak out. Speaking could mean saying an answer, sharing a strategy, asking a question of a specific classmate, or asking a question to the class in general.

The next thing that we did was work on the repeating decimals to fraction algorithm. This is a challenge for most people and if I were to venture a guess I'd say more than 50% of college educated adults could not do this. Thus we struggled. That said, students successfully were able to take the decimal form of fractions as they are presented in the calculator image above and correctly remove the rounded portion of the decimal. Where the conversation baffled them was in two places. First of all, the idea that a number such as 0.16 with the six repeating (apologies on not showing that notation in Blogger) was not equivalent to 16 when multiplied by 100. For some students recognizing what happened to a number multiplied by 100 was an issue.

The second issue came when the subtraction of 100x - 10x was complete and students had to take the 90x remaining and set it equal to 15 (the difference of 16.6 repeating and 1.6 repeating). Students were doing 90 divided by 15 instead of 15 divided by 90. I was quick to ask how a number like 0.16 could be greater than one as students told me the answer was 6.

In terms of where we go from here, I think there is a certain degree of repetition that will need to be revisited here, but also getting students familiar with the various ways to write a repeating decimal and what that number looks like multiplied by various bases of 10.

Lesson Objective (Honors Math): Solve for a variable in a multiple step equation

Lesson Sequence: I read the quote and question, but accidentally skipped the number talk. It was probably for the better as the time we spent on two-step equations was valuable.

I gave students the problem 2/3p - 4 (p + 3) = -7p + 10 as a My Favorite No problem. Index cards were distributed for each student. I gave students about seven minutes to work out the problem (the students must have thought it seemed more like one minute because many were stumped). Here were some of the answers that I saw.

This was probably the most common error. As it was displayed on the board, a couple students were quick to point out the error that there should be a minus 12 instead of a plus 12. I tried to help them by stating that the distributive property is multiplication and we are really "taking away four groups of p +3." 

Here the student did almost everything perfectly until the last step. The 22 to the left of the equation was ignored and the 7p that was added to the left side at the end never got added back to the right side. Too often I am guilty of giving credit for right and no credit for wrong. Upon showing work like this to students I am not sure if that's fail. It's fair to say that the student has work to do in terms of mastering the standard, but by the same token the student can distribute correctly and also knows what is needed to balance equations. 

This is a third positive example that does not demonstrate mastery, but is certainly close to achieving that. The student writes that -12 - 10 is -2. Perhaps this is a complete misunderstanding of that concept or just a simple mistake in this one instant. How is that measured? At the end there is also some difficulty combining like terms, but this only appears to be the case with the fractional coefficient. 

After the My Favorite No, I had students work in groups of three on problems that were placed throughout the room to get them out of their seats. Through working in groups of three, they were able to get feedback on these minor issues that I might not have been able to give them if they were working individually. At the end of class I asked students what they would consider thinking about as they were doing a two-step equation to make sure they did not make a mistake that they had made earlier in class.